Input-Output Models y = Hu u(t) is a piecewise continuous function of t and
belongs to a linear space of signals. The space of bounded functions: sup t≥0 u(t
)
Nonlinear Systems and Control Lecture # 20 Input-Output Stability
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Input-Output Models y = Hu u(t) is a piecewise continuous function of t and belongs to a linear space of signals
The space of bounded functions: supt≥0 ku(t)k < ∞ The space of square-integrable functions: R∞ T 0 u (t)u(t) dt < ∞
Norm of a signal kuk:
kuk ≥ 0 and kuk = 0 ⇔ u = 0 kauk = akuk for any a > 0
Triangle Inequality: ku1 + u2 k ≤ ku1 k + ku2 k
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Lp spaces: L∞ : kukL∞ = sup ku(t)k < ∞ t≥0
L2 ; kukL2 = Lp ; kukLp =
Z
∞ 0
sZ
∞ 0
uT (t)u(t) dt < ∞
ku(t)kp dt
1/p
< ∞, 1 ≤ p < ∞
Notation Lm p : p is the type of p-norm used to define the space and m is the dimension of u
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Extended Space: Le = {u | uτ ∈ L, ( ∀ τ ∈ [0, ∞)} u(t), 0 ≤ t ≤ τ uτ is a truncation of u: uτ (t) = 0, t>τ
Example:
Le is a linear space and L ⊂ Le ( t, 0 ≤ t ≤ τ u(t) = t, uτ (t) = 0, t>τ u∈ / L∞ but uτ ∈ L∞e
q is causal if the value Causality: A mapping H : Lm → L e e of the output (Hu)(t) at any time t depends only on the values of the input up to time t
(Hu)τ = (Huτ )τ
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q is L stable if ∃ α ∈ K Definition: A mapping H : Lm → L e e β ≥ 0 such that
k(Hu)τ kL ≤ α (kuτ kL ) + β,
∀ u ∈ Lm e and τ ∈ [0, ∞)
It is finite-gain L stable if ∃ γ ≥ 0 and β ≥ 0 such that k(Hu)τ kL ≤ γkuτ kL + β,
∀ u ∈ Lm e and τ ∈ [0, ∞)
It is small-signal L stable (respectively, finite-gain L stable) if ∃ r > 0 such that the inequality is satisfied for all u ∈ Lm e with sup0≤t≤τ ku(t)k ≤ r
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Example: Memoryless function y = h(u) h(u) = a + b tanh cu = a + b ′
h (u) =
4bc (ecu +
2 e−cu )
ecu − e−cu ecu
+
e−cu
,
a, b, c > 0
≤ bc ⇒ |h(u)| ≤ a+bc|u|, ∀ u ∈ R
Finite-gain L∞ stable with β = a and γ = bc h(u) = b tanh cu,
|h(u)| ≤ bc|u|, ∀ u ∈ R Z ∞ Z ∞ |h(u(t))|p dt ≤ (bc)p |u(t)|p dt, for p ∈ [1, ∞) 0
0
Finite-gain Lp stable with β = 0 and γ = bc
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h(u) = u2 sup |h(u(t))| ≤ t≥0
sup |u(t)| t≥0
!2
L∞ stable with β = 0 and α(r) = r 2
It is not finite-gain L∞ stable. Why? h(u) = tan u |u| ≤ r
0 such that for each x(0) with kx(0)k < k1 , the system x˙ = f (x, u),
y = h(x, u)
is small-signal L∞ stable
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Example x˙ 1 = −x31 + x2 ,
x˙ 2 = −x1 − x32 + u,
y = x1 + x2
V = (x21 + x22 ) ⇒ V˙ = −2x41 − 2x42 + 2x2 u x41 + x42 ≥ 21 kxk4 V˙
≤ −kxk4 + 2kxk|u| = −(1 − θ)kxk4 − θkxk4 + 2kxk|u|, 0 < θ < 1 1/3 ≤ −(1 − θ)kxk4 , ∀ kxk ≥ 2|u| θ
ISS
L∞ stable
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